1
Group Frames with Few Distinct Inner Products and Low Coherence
arXiv:1509.05087v1 [cs.IT] 17 Sep 2015
Matthew Thill Babak Hassibi Department of Electrical Engineering, Caltech, Pasadena, CA
Abstract—Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. Highperformance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original “group frame” design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian’s original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with n elements from n(n−1) to n − 1. All of our constructions 2 further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When n is prime, for example, we use cyclic groups to construct m-dimensional distinct inner products. frame vectors with at most n−1 m We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance. Index Terms—Coherence, frame, unit norm tight frame, group representation, group frame, Welch bound, spherical codes, compressive sensing.
I. Introduction: Frames and Coherence Recall that a frame is the following generalization for the basis of a vector space: Definition 1: Let V be a vector space equipped with an inner product h·, ·i (or more specifically, a separable Hilbert space). A set of elements {fk }k∈I , where I is a countable index set, is a frame for V if there exist positive Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to
[email protected]. Email: {mthill,hassibi}@caltech.edu. This work was supported in part by the National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from ˜ Jet Propulsion Laboratory through the Qualcomm Inc., by NASAOs ˜ Fund, by King Abdulaziz University, and President and DirectorOs by King Abdullah University of Science and Technology.
constants A and B such that X A||f ||22 ≤ |hf, fk i|2 ≤ B||f ||22 ,
(1)
k∈I
for all f ∈ V. A frame is called tight if A = B in this definition, and unit norm if ||fk ||2 = 1, ∀k ∈ I. We define the coherence µ of the frame to be the maximum correlation between any two distinct columns: µ = max i6=j
|hfi , fj i| . ||fi ||2 · ||fj ||2
Designing frames with low coherence is a problem that has connections to a wide range of fields, including compressive sensing [7]–[9], [25], [26], [56], spherical codes [21], [50], LDPC codes [28], MIMO communications [33], [34], quantum measurements [27], [44], [46], etc. Frame theory has also made its mark as an interesting field in its own right, with a great collection of recent work by Casazza, Kutyniok, Fickus, Mixon, and many others [10], [11], [13], [15], [16], [29]. Most often we will consider our frame vectors to be the columns {mi }ni=1 of a matrix M = [m1 , m2 , . . . , mn ] ∈ Cm×n . We will speak of the coherence of M to be the coherence of the frame {mi }. The frame is tight if and only if MM∗ = λIm where Im is the m×m identity matrix and λ is a scalar. Furthermore, λ = A = B in (1). If the {mi } n . form a unit norm tight frame, then λ = m It is easy to see that any orthonormal basis for a vector space is a tight frame, and consequently a frame can be regarded as a generalization of an orthonormal set which may include more vectors than the dimension of the space. One important example of a tight frame that we will encounter is when the rows of M are a subset of the rows of the n × n discrete Fourier matrix: 2πi Definition 2: Let ω = e n , and let F be the discrete Fourier matrix, whose (i, j)th entry is ω ij . Let M = [m1 , ..., mn ] ∈ Cm×n such that the rows of M are a subset of the rows of λF for some scalar λ ∈ C. Then {mi }ni=1 is called a harmonic frame. Remark: The notion of a harmonic frame is actually more general than in this definition, as is explained in [14], but the more general harmonic frames are also tight with equal norm elements. For our purposes, the above definition will suffice. We will touch on generalized harmonic frames in Section IV, but there is a substantial collection of results on harmonic frames in [17] and [32]. Of great interest is when a tight frame is equiangular :
2
Definition 3: A unit-norm frame {fk }k∈I is said to be equiangular if there is some constant α such that for any i 6= j, |hfi , fj i| = α. The following theorem, known as the Welch bound and based on the results of [63], provides a lower bound on the coherence of a frame: Theorem 1: Let E be the field of real or complex numbers, and {fk }nk=1 be a unit-norm frame for Em . Then r n−m , (2) max |hfi , fj i| ≥ i6=j m(n − 1) with equality if and only if {fk }nk=1 is tight and equiangular. Proof: This theorem is a classical result, and one of an even broader set of bounds [63]. A quick proof which can be found in [50] involves considering the eigenvalues of the Gram matrix G defined by Gij = hfi , fj i.
same number of times. Then the coherence µ of {fi } is at √ most a factor of r greater than the Welch bound. That is, r √ n−m . (3) µ≤ r m(n − 1) Proof: As a preliminary fact, Theorem 6.2 of [4] shows P 2 that the frame potential i,j |hfi , fj i|2 is at least nm with equality if and only if the frame is tight. Let α1 , ..., αr be the distinct squared absolute values of the inner products, {|hfi , fj i|2 }i6=j . Since each of the αi occurs the same number of times as a squared inner product norm, we have that their arithmetic mean is equal to that of the {|hfi , fj i|2 }i6=j , which is r
X 1 n−m 1X αi = |hfi , fj i|2 = , r i=1 n(n − 1) m(n − 1)
(4)
i6=j
Thus, we would like to identify tight, equiangular frames for use in constructing matrices which achieve this lower bound. This problem arises in various contexts, for example line packing problems [19]. It should be emphasized that such frames do not exist for all values of m and n, so in general, we would also like to find ways to optimize the coherence by choosing M wisely from a cleverly designed class of matrices. Our approach will be to use the group frame construction proposed by Slepian [48] in the 1960s. Group frames have received a great deal of attention in recent years, notably in the substantial collection of work by Vale, Waldron, and others [17], [32], [58]–[60], [62]. For an excellent review of the work in group frames, see [14]. On one final note before proceeding, a common approach to produce a set of vectors with low correlation is to construct a set of Mutually Unbiased Bases (MUBs). Two bases {e1 , ..., em } and {e01 , ..., e0m } for Cd are mutually unbiased if each is orthonormal, and |hei , e0j i| = √1m for any i and j. Algebraic constructions of up to m + 1 MUBs are known in prime-power dimensions m, allowing for a number of vectors at most m2 + m [2], [41], [64]. The frame constructions presented in this paper will at times outperform this coherence, though typically with a smaller number of vectors. More importantly, though, our frames do not require m to be prime. II. Reducing the Number of Distinct Inner Products in Tight Frames In practice, constructing frames which are both tight and equiangular can prove difficult. It turns out, however, that we can expect reasonably low coherence from tight frames if we just require that the inner products between frame elements take on few distinct values, provided that each of these values arises the same number of times. The following lemma, which is in some sense a generalization of the Welch bound, provides a bound on the coherence of a tight frame: Lemma 1: Let {fi }ni=1 ⊂ Cm be a unit-norm tight frame such that the absolute values of the inner products, |hfi , fj i|i6=j , take on r distinct values, each occurring the
where the second equality follows from the preliminary frame potential result and the fact {fi }ni=1 is tight and unit-norm by assumption. Thus, since all the αi are nonnegative we see that 2
µ = max αi ≤ i
r X i=1
αi = r ·
n−m , m(n − 1)
(5)
from which the result follows. In light of Lemma 1, our goal will be to construct a tight frame whose elements have very few inner product values between them, each of which occurs with the same multiplicity. In the following sections, we will present a group theoretic way to do this. III. Frames from Unitary Group Representations: Slepian Group Codes In [48], Slepian proposed a method to construct lowcoherence matrices by reasoning that the key to controlling the inner products between the columns was to reduce the number of distinct inner product values which arise. His construction, which has come to be known as a group frame, has since been generalized (see, for example [59] and [14]). On this note, let U = {U1 , U2 , ..., Un } be a (multiplicative) group of unitary matrices. We can equivalently view U as the image of a faithful, unitary representation of a group G. In some works, e.g. [30], U is taken to be a group-like unitary operator system—the image of a projective representation—but normal representations will suffice for our purposes. Such representations exist for any finite group. Suppose that for each i, we have Ui ∈ Cm×m (or equivalently, U is the image of an m-dimensional representation). Let v = [v1 , ..., vm ]T ∈ Cm×1 be any vector, and let M be the matrix whose ith column is Ui v: M = [U1 v, ..., Un v]. The inner product between the ith and j th columns of M is hUi v, Uj vi = v∗ U∗i Uj v. Since U is a unitary group, we have U∗i Uj = U−1 i Uj = Uk , for some k ∈ {1, ..., n}, so we can write hUi v, Uj vi = v∗ Uk v. In this manner, we have reduced the total number of pairwise inner products
3
between the columns of M from n2 to n − 1, the inner products parametrized by the non-identity elements of U. Furthermore, we have the following: Lemma 2: Let {U1 , ..., Un } ⊂ Cm×m be a set of distinct unitary matrices which form a group under multiplication, and let v ∈ Cm×1 be a nonzero vector. Each of the values v∗ Uk v occurs as the inner product between two columns of M = [U1 v, ..., Un v] the same number of times. Proof: For every choice of Uk and Ui , there is a unique Uj such that U−1 i Uj = Uk . Thus, for each Uk , there are n pairs (Ui , Uj ) such that v∗ U∗i Uj v = v∗ Uk v.
the frame matrix M = [v, Uv, ..., Un−1 v]. Note that if we express our group elements diagonally as in (6),
IV. Abelian Groups and Harmonic Frames
If the ki are distinct this is a subset of rows of the discrete Fourier matrix, hence a harmonic frame. For cyclic groups, the inner product between the columns U`1 v and U`2 v, after normalizing the columns, ∗ `2 −`1 will take the form |v U||v||2 v| , which is the value of the 2 inner product determined by U`2 −`1 in (8). A general abelian group G can be represented as follows: First express G as a direct product of, say, L cyclic groups of orders n1 , ..., nL , so that G ∼ = nZ1 Z × ... × nLZ Z . Then let ω1 , ..., ωL be the corresponding primitive roots of unity: k k ωj = e2πi/nj . Then we set Uj = diag(ωj 1j , ..., ωj mj ), where we will assume that the kij are distinct integers modulo nj . The abelian group generated by the diagonal matrices {U1 , ..., UL } is isomorphic to G, and an arbitrary element will take the form Ua1 1 Ua2 2 . . . UaLL , where aj ∈ {0, ..., nj − 1}. Our frame matrix M will then take the form M = [. . . (Ua1 1 Ua2 2 . . . UaLL v) . . .]0≤aj ≤nj −1 . In this form, our previous cyclic frames clearly arise as subsets of the columns of M. It turns out that these abelian frames are the generalized harmonic frames as described in [14], up to a unitary rotation of v or an equivalent group representation. The frame matrix M is a subset of rows h of the Kroneckeri product A1 ⊗ ... ⊗ AL , n −1 where Aj = v, Uj v, ..., Uj j v . As such, any of these frames will be tight.
For now, we will restrict ourselves to consider representations of abelian groups. Abelian groups are the simplest groups, in a sense, and have the special property that each of their irreducible representations is one-dimensional. Therefore, if U is the image of a representation of an abelian group, then all of the elements Ui can be simultaneously diagonalized by a change of basis matrix. Thus, we may assume without loss of generality that the Ui are diagonal unitary matrices whose diagonal entries are 2πi powers of ω = e n : Uj = diag(ω k1,j , ..., ω km,j ) ∈ Cm×m ,
(6)
where the ki,j are integers. With each Uj in this form, the inner products between the normalized columns of M will take the form m 2 |v∗ Uj v| X |vi | ki,j = ω , ||v||22 ||v||22 i=1
T
where v = [v1 , ..., vm ] . So we see that the entries of v simply weight the diagonal entries of Uj in the above sum. In particular, without loss of generality, we may take the entries of v to be real. Furthermore, it turns out that in order for our abelian group frame to be tight, all the entries vi must be of equal norm. This follows from Theorem 5.4 in [14]. On this note, we will consider the case where v is the vector of all 1’s, v = 1m = [1, ..., 1]T ∈ Cm×1 ,
(7)
so that the above inner product norm becomes simply m |v∗ Uj v| 1 X ki,j = ω . (8) ||v||22 m i=1
Notice that from Equation (8), we can see that the coherence of our final matrix would remain unchanged if we chose ω to be any other primitive nth root of unity. Let us examine the simple case where U is a cyclic unitary group, the most basic abelian group. That is, the elements of U can be written as the powers of a single unitary matrix U of order n: U = {U, U2 , ..., Un−1 , Un = Im }, where Im is the m × m identity matrix. We consider again choosing v to be the vector of all ones as in (7), and form
U = diag(ω k1 , ..., ω km ) ∈ Cm×m , we can see that the matrix M will take the form M = v Uv . . . Un−1 v 1 ω k1 ω k1 ·2 . . . ω k1 ·(n−1) 1 ω k2 ω k2 ·2 . . . ω k2 ·(n−1) = . . .. .. .. .. .. . . . . 1
ω km
ω km ·2
...
(9)
(10)
(11)
ω km ·(n−1)
V. Equiangular Frames from Cyclic Group Representations Let us examine the harmonic frame formed by the columns of M in (11). [65] classified the conditions on the ki under which this frame is equiangular. Since we know these frames are tight, this determines precisely when their coherence achieves the Welch lower bound of Theorem 1. Definition 4: Let G be a group. A difference set K = {k1 , ..., km } ⊂ G is a set of elements such that every nonidentity element g ∈ G occurs as a difference ki −kj the same number of times. That is, the sets Ag := {(ki , kj ) ∈ K × K | ki − kj = g} have the same size for g 6= 0. Theorem 2 ( [65] Equiangular Harmonic Frames): The harmonic frame formed by the columns of M in (11) is equiangular if and only if the integers ki form a difference set in Z/nZ. Proof: The proof follows from a simple but insightful Fourier connection. Let us define At := {(ki , kj ) ∈ K × K | ki − kj ≡ t mod n} for any t ∈ Z/nZ, and
4
set at := |At |. Furthermore, if we index the columns as ` = 0, 1, ..., n − 1 then the inner product associated to the `th column takes the form 1 X `k v ∗ U` v = ω . c` := 2 ||v||2 m k∈K
Since we are concerned only with the magnitude of c` , we may consider the quantity !∗ ! X X 1 `k `k 2 ω ω α` := |c` | = 2 m k∈K
=
1 m2
X
k∈K
ω `(ki −kj ) .
ki ,kj ∈K
We can then write α` =
n−1 1 X at ω `t , m2 t=0
(12)
which gives us a Fourier pairing between the α` and the at with inverse transform given by at =
n−1 m2 X α` ω −t` . n
(13)
`=0
M will be an equiangular tight frame precisely when all of the α` are equal for ` 6= 0, and from the Fourier pairing this will occur precisely when the at are equal for t 6= 0, i.e., when the ki form a difference set. This concept of tight equiangular frames arising from difference sets has since been generalized and elaborated [23], [14], [62]. [22] showed how slightly relaxed forms of difference sets can produce frames which have coherence almost reaching the Welch Bound. Many of our results in the following sections can also be viewed as using more extensively-relaxed difference sets to produce lowcoherence frames. Difference sets have been long studied and classified [3], [5]. They have found application in other fields as well, such as designing codes for DS-CDMA systems [24], LDPC codes [61], sonar and synchronization [31], and other forms of frame design [38]. While Theorem 2 completely characterizes the optimalcoherence frames arising from representations of cyclic groups, it reveals that equiangular frames of the form (11) are rather scarce, since the number of known difference sets is relatively small. In the following section, we will present a new strategy for selecting the integers ki which, while not always producing an equiangular frame, does yield frames with few distinct inner product values and provable low coherence. VI. Cyclic Groups of Prime Order We have already managed to cut down the number of distinct inner products between columns from n2 to n−1, simply by using a unitary group to generate our columns. For cyclic groups, however, we can reduce this number even more. We first consider the case where n is prime. × Let G = (Z/nZ) , the multiplicative group of the integers
modulo n. As usual, we identify the elements of Z/nZ with the integers 0, 1, ..., n−1. Since n is assumed to be a prime, G is itself a cyclic group, and consists of the n − 1 nonzero elements of G. Now let us choose m to be any divisor of n − 1, and set r := n−1 m . Since G is cyclic, it has a unique subgroup K of order m consisting of the distinct rth powers of the elements of G. In fact, if g is any generator n−1 for G, then K will be generated by k := g m . Now, if we write out the elements of K as {k1 , ..., km } (or equivalently in terms of a single generator k as {1, k, k 2 , ..., k m−1 }), we can form our generator matrix U as in (9), choosing ω ki to be the ith diagonal term. Note that since K consists of elements relatively prime to n, then for each i, ω ki has multiplicative order n. It follows that U also has order n n−1 ∼ and generates the cyclic group U = {U` }`=0 = Z/nZ. It turns out that this construction not only reduces the number of distinct inner product values between our columns, but it maintains the property that each such value occurs with the same multiplicity: Theorem 3: Let n be a prime and m any divisor of n − 1. Take K = {k1 , ..., km } to be the unique (cyclic) 2πi subgroup of G = (Z/nZ)× of size m. Set ω = e n , v = √1m [1, ..., 1]T ∈ Rm , and U = diag(ω k1 , ..., ω km ). Then the columns of M = [v, Uv, ..., Un−1 v] form a unitnorm tight frame with at most n−1 m distinct inner product values between its columns, each occurring with the same multiplicity. Proof: For any integer ` in the set {1, ..., n − 1}, the inner product corresponding to U` (as in Equation (8)) will take the following form: m |v∗ U` v| 1 X `·ki = ω . ||v||22 m i=1
(14)
Notice the exponents of ω appearing in the above summation can be taken modulo n, since ω is an nth root of unity, and are then simply the elements of the `th coset of K in G, `K = {` · k1 , ..., ` · km }. The set of all cosets of K in G is denoted G/K. From elementary group theory, we know that the distinct cosets of K form a disjoint partition of G, so the number of distinct cosets of K in G is the |G| = n−1 quotient of their sizes: |G/K| = |K| m . Thus, the total number of distinct pairwise inner products that we now must control is n−1 m . It only remains to show that each of the n−1 m inner products occurs the same number of times. Let {`1 , ..., `r } be a complete set of coset representatives for K in (Z/nZ)× . Here, r is simply n−1 m . Then every element in {1, ..., n − 1} can be written uniquely as a product `i kj , and from Lemma 2 the n − 1 inner products v∗ U`i kj v all arise the same number of times. As described above, the n−1 m distinct inner product values correspond to the cosets of K, i.e., for a fixed `i the m inner products v∗ U`i k1 v, ..., v∗ U`i km v will give rise to one of the distinct inner product values. Thus, since each distinct value corresponds to m inner products, each arising the same number of times, our result is proved.
5
TABLE I Coherences for Random and Group Matrices (for n a Prime) (n, m) (251, 125) (499, 166) (499, 249) (503, 251) (521, 260) (521, 130) (643, 321) (643, 214) (701, 175) (701, 350) (1009, 504) (1009, 336) (1009, 252)
Complex Gaussian .2677 .3559 .2226 .2137 .2208 .3065 .2034 .2274 .2653 .1788 .1565 .2086 .2287
Random Fourier .1996 .1786 .1736 .1533 .1504 .2376 .1627 .1978 .2316 .1326 .1147 .1384 .1631
Group Matrix .0635 .0888 .0449 .0447 .0458 .1175 .0395 .0755 .0687 .0393 .0325 .0597 .0846
q
n−m m(n−1)
.0635 .0635 .0449 .0447 .0439 .0761 .0395 .0559 .0655 .0379 .0315 .0446 .0546
VII. Sharper Bounds on Coherence for Frames from Cyclic Groups of Prime Order In the special case where we construct our frame as in Theorem 3 (using Slepian’s approach with a group U ∼ = Z/nZ and n prime), we have a great deal of underlying algebraic structure in our frame. So it should come as no surprise that we can derive sharper bounds on our coherence and even compute it exactly in some cases. As before, let m be a divisor of n − 1, and take K = {k1 , ..., km } to be the unique subgroup of (Z/nZ)× of size m. Define r := n−1 m , which is the number of distinct inner product values. If r is small, it becomes relatively simple to analyze these values. For example: Theorem 4 (r = 2): Let n be a prime, m a divisor of 2πi n − 1, and ω = e n . Let K = {k1 , ..., km } be the unique subgroup of (Z/nZ)× of size m, and set U = diag(ω k1 , ..., ω km ) ∈ Cm×m , v = √1m [1, ..., 1]T ∈ Cm×1 , and M = [v, Uv, ..., Un−1 v]. If r := n−1 m = 2, there are two distinct inner product values between the columns of M, both of which are real. If
Norm of Inner Product
(a) 0.2 0.15 0.1 0.05 0
0
100
200
300
400
500
200 300 400 Group Element (Power of U)
500
(b) Norm of Inner Product
We emphasize the power of this construction in reducing the number of inner products that we must control in order to maintain low matrix coherence. Since we are free to choose m to be any divisor of n − 1, then for properly chosen matrix dimensions, we can reasonably create matrices with just two or three distinct values of inner products between columns. In practice, this often creates matrices with remarkably low coherence, far outmatching that of any known randomly-generated matrices. In Table I, we compare the coherences of the “Group Matrices” from our construction with those of randomly-generated complex Gaussian matrices and matrices designed by randomly selecting m rows from the n × n Fourier matrix. (This latter construction is equivalent to randomly selecting the exponents ki in our cyclic generator matrix U in (9).) For convenience, we also list the lower bound on coherence from Theorem 1, and we underline the coherences which achieve this bound. Figure 1 illustrates explicitly the inner products for a random Fourier matrix vs. a Group matrix.
0.2 0.15 0.1 0.05 0
0
100
Fig. 1. The norms of the inner products associated to each group element for (a) randomly-chosen K, and (b) K selected to be a subgroup of (Z/pZ)× of index 3. Here, n = p = 499, m = 166. In (b), as expected, there are only three distinct values of the inner products between distinct, normalized columns.
n−1 is divisible by 4, these inner q products are n−m− 1
√ −1± 1+2m . 2m
1 2 In this case, M has coherence m(n−1) + 2m . If n − 1 is not divisible by 4, then the columns of M form frame. The two inner products are q an equiangular q 1 1 1 n−m ± m 2 + 2m , and the coherence is m(n−1) . Proof: We will hold off on the details of the proof until Appendix B aside from mentioning that it is related to the connection made by Xia et al [65] between tight equiangular harmonic frames and difference sets. In fact, in the case where n − 1 is not divisible by 4, K forms a known difference set in Z/nZ. If we view Z/nZ as the additive group of Fn , this particular case also overlaps with the tight equiangular frames classified in Theorem 3 of [23].
As the number r of inner products increases, it becomes more complicated to explicitly compute their values or even just the coherence of the resulting frame. While there were only two cases to consider when r = 2, there are many more even for r as low as 3. We can, however, exploit the algebraic structure of our frames to yield bounds on their coherence which in practice prove to be nearly tight. Theorem 5 (r = 3): Let n be a prime, m a divisor of 2πi n − 1, and ω = e n . Let K = {k1 , ..., km } be the unique subgroup of (Z/nZ)× of size m, and set U = diag(ω k1 , ..., ω km ) ∈ Cm×m , v = √1m [1, ..., 1]T ∈ Cm×1 , and M = [v, Uv, ..., Un−1 v]. If r := n−1 m = 3, then the coherence of M will satisfy s ! r 1 1 1 1 4 3+ + µ≤ 2 ≈ , (15) 3 m m m 3m
6
and for large enough m, we will asymptotically have the following lower bound on coherence: (16)
which is strictly greater than the Welch bound. Proof: We present the proof in Appendix C. From Theorem 5 we see that unlike when r = 2, we can never hope to achieve the Welch bound with these frames when r = 3. But this is not a trend, for our frames will again be able to achieve the Welch bound for certain higher values of r, including r = 4 and r = 8. This again relates to the connection with difference sets from [65]. As a result, the lower bound on coherence in Theorem 5 does not generalize to all values of r. Fortunately, the upper bound does: Theorem 6 (General r): Let n be a prime, m a divisor 2πi of n − 1, and ω = e n . Let K = {k1 , ..., km } be the unique subgroup of (Z/nZ)× of size m, and set U = diag(ω k1 , ..., ω km ) ∈ Cm×m , v = √1m [1, ..., 1]T ∈ Cm×1 , and M = [v, Uv, ..., Un−1 v]. If r := n−1 m , then the coherence µ of M satisfies the following upper bound: s ! 1 1 1 1 µ≤ (r − 1) r+ + . (17) r m m m
0.2
Coherence
1 µ ≥ √ (asymptotically), m
Upper Bound Lower Bound Welch Bound Actual Coherence
0.25
0.15
0.1
0.05
0 0
50
100
150
200
250
m
300
400
450
500
Fig. 2. The upper and lower bounds on coherence for r = 3.
Upper Bound (m even) Lower Bound (m even) Upper Bound (m odd) Lower Bound (m odd) Welch Bound Actual Coherence (m even) Actual Coherence (m odd)
0.3
Coherence
0.25
Proof: This theorem will be proved in Appendix D. This bound is strictly lower than the one from Lemma 1, which applies to all tight frames. In fact, when n > 2, we can find an even lower bound on the coherence of our frames constructed in Theorem 3 which surprisingly depends only on whether m is odd: Theorem 7 (m odd): Let n be an odd prime, m a divisor 2πi of n − 1, and ω = e n . Let K = {k1 , ..., km } be the unique subgroup of (Z/nZ)× of size m, and set U = diag(ω k1 , ..., ω km ) ∈ Cm×m , v = √1m [1, ..., 1]T ∈ Cm×1 , and M = [v, Uv, ..., Un−1 v]. Set r := n−1 m . If m is odd, then the coherence of M is upper-bounded by s r 2 r 2 1 1 µ≤ + −1 β + β2, (18) r m 2 2 q 1 1 where β = m r+ m . Proof: We delay the proof of this theorem until Appendix D. It is worth noting that this latter bound has no analog in the r = 3 situation because m must always be even in that case. We explain the reason for this in the sequel, and give an alternate classification for exactly when this latter coherence bound applies. We illustrate the upper and lower bounds for r = 3 in Figure 2 and the two upper upper bounds from Theorem 7 for when r = 4. When r = 4, we can also derive different lower bounds on the coherence for when m is even or odd, and together with the two upper bounds from the theorems they form two non-overlapping regions in which the coherences can fall
350
0.2
0.15
0.1
0.05
0 0
50
100
150
200
250
m
300
350
400
450
500
Fig. 3. The upper and lower bounds on coherence for r = 4.
in the graph. While these regions will exist for every r, they will sometimes overlap (that is, the lower bound on coherence for m even could be less than the upper bound for m odd). VIII. Generalized Dihedral Groups Rather than dwell on clever constructions of general abelian groups, let us instead investigate what changes when U is nonabelian. In this case the irreducible representations at our disposal will no longer all be onedimensional, so we will no longer have all the matrices Ui be simultaneously diagonal. Consequently, it is no longer clear that we can restrict our vector v to be real-valued.
7
One simple class of nonabelian groups is that of semidirect products of cyclic groups. On this note, consider the following group presentation (which arises in [47]): Gn,r = hσ, τ | σ n = 1, τ D = 1, τ στ −1 = σ r i.
(19)
Here, n and r are relatively prime integers, and D is the multiplicative order of r modulo n. Gn,r is precisely a Z Z o DZ , and if we take semidirect product in the form nZ D = 2 and r = n − 1, we see that we obtain the familiar dihedral group D2n . There are n·D group elements in Gn,r , each of which can be written in the form σ a τ b for some integers 0 ≤ a < n and 0 ≤ b < D. Gn,r has an irreducible representation in the form ω ωr σ 7→ S := (20) ∈ CD×D , .. . ωr τ→ 7 T := 1
1 ..
.
where the indices are taken modulo D. It turns out that we can satisfy all our restrictions on w by selecting its indices to form a Zadoff-Chu (ZC) sequence [18], [37]:
wd =
∈ CD×D , 1
(21)
2πi
where ω = e n (see again [47]). The informed reader might note that this representation is quite similar to that of Heisenberg groups, which have been extensively applied to the construction of frames [6], [40], [44], [45]. Our following methods can be conceivably adjusted for use with Heisenberg frames as well. In order to construct our frames, we would like to follow the example of our previous construction in Theorem 3 by selecting a representation for Gn,r of the form k S 1 T .. .. σ 7→ [σ] := , , τ 7→ [τ ] := . . Skm
d
D−1
1
T vector v = wT wT ... wT ∈ CDm×1 . The question now becomes how to choose w? In order to preserve as much of the structure from our previous construction as possible, we would like each entry of w to have the same norm. This will ensure that the inner products corresponding to the elements [σ]a will have the same values as those in our previous construction from Theorem 3 corresponding to when U was the cyclic group Z/nZ generated by [σ]. Let us require that wd be unit norm for each d, and consider attempting to force w to satisfy the constraint that X w∗ Tb w = wd∗ wd+b = 0, ∀b (23)
T (22)
where m and the ki are cleverly chosen integers. Then we will select a vector v ∈ CDm×1 and take our frame to be the columns of the matrix M := [. . . [σ]a [τ ]b v . . .]0≤a
